Place the quartered beets and 3-4 uncooked eggs in the bottom of a small saucepan.
Cover with enough cool water to cover the eggs by about 1 inch, and add the vinegar.
Bring to a boil over high heat; remove from heat, cover, and set a timer for 15 minutes.
At the end of the 15 minutes, remove the lid, place the eggs in a bowl and strain the dye into the bowl.
Place the bowl in the refrigerator to cool.
Leave the eggs in the dye for as little as one hour or as much as 12, until desired color hue is achieved.
Remove eggs from the dye with a slotted spoon and place on a rack in the fridge to dry.